BGG resolution

example of BGG resolution : $\mathfrak{sl}_2$

• \(W_{\lambda}\) : irreducible highest weight module
• \(V_{\lambda}\) : Verma modules
  • note that the Verma modules are free modules of rank 1 over \(\mathbb{C}[F]\) where $F$ is the annihilation operator of $\mathfrak{sl}_2$
• \(\lambda ,-2+\lambda ,\cdots, -\lambda, -\lambda-2,\cdots\)
• \(W_{\lambda}=V_{\lambda}/V_{-\lambda-2}\)
• BGG resolution\(0\to V_{-\lambda-2}\to V_{\lambda}\to W\to 0\)
• number of modules = 2 (=order of Weyl group in general)
• character of W = alternating sum of characters of Verma modules\(\chi_{W_{\lambda}}=\chi_{V_{\lambda}}-\chi_{V_{-\lambda-2}}=\frac{q^{\lambda}}{1-q^{-2}}-\frac{q^{-\lambda-2}}{1-q^{-2}}\)
• comparison with Weyl-Kac character formula

\[ch(W_{\lambda})=\frac{\sum_{w\in W} (-1)^{\ell(w)}w(e^{\lambda+\rho})}{e^{\rho}\prod_{\alpha>0}(1-e^{-\alpha})}=\frac{q^{\lambda+1}-q^{-\lambda-1}}{q^{1}(1-q^{-2})}\] where I used \(\rho=1,\alpha=2\) and \(w(\lambda+\rho)=-\lambda-\rho\)

maps between Verma modules

• 2 conditions to have non-zero homomorphisms \(V_{\lambda}\to V_{\mu}\) between two Verma modules
  • \(\lambda+\rho, \mu+\rho\) are in the same orbit of Weyl group
  • \(V_{\lambda}\leq V_{\mu}\), i.e. \(\lambda = \mu -\sum \alpha\), where the sum is over some positive roots.
• example in SL2
  • \(\lambda = \mu -2n\), \(n=0,1,2,\cdots\)
  • \((\lambda+1)^2 = (\mu+1)^2\)

books

• James E. Humphreys, Representations of Semisimple Lie Algebras in the BGG Category O, Grad. Stud. Math., 94, Amer. Math. Soc., Providence, RI, 2008.